Fuzzy ELECTRE I (Group, Trapezoidal)

1. 1. Form the group of K DMs and determine evaluation criteria (m alternatives, n criteria).

   $$E=\{e_1,e_2,\dots ,e_K\},A=\{A_1,\dots ,A_m\},C=\{C_1,\dots ,C_n\}$$

2. 2. Each DM k assigns linguistic performance ratings x̃_ijk to action A_i on criterion C_j.

   $${\tilde{x}}_{ijk}=(x_{ijk}^l,x_{ijk}^p,x_{ijk}^q,x_{ijk}^u),i=\mathrm{1..}m,j=\mathrm{1..}n,k=\mathrm{1..}K$$

3. 3. Each DM k assigns linguistic weights w̃_jk to each criterion C_j.

   $${\tilde{w}}_{jk}=(w_{jk}^l,w_{jk}^p,w_{jk}^q,w_{jk}^u),j=\mathrm{1..}n,k=\mathrm{1..}K$$

4. 4. Convert linguistic evaluations into trapezoidal fuzzy numbers via the predefined linguistic scale (Tables 1-2, Figs. 4-5).

   $$linguistic\_term\mapsto (l,p,q,u)$$

5. 5. Aggregate K DMs' ratings and weights using min-mean-mean-max scheme (Eqs.5-8).

   $${\tilde{x}}_{ij}=(x_{ij}^l,x_{ij}^p,x_{ij}^q,x_{ij}^u),x_{ij}^l=\underset{k}{\min }{x}_{ijk}^l,x_{ij}^p=\frac{1}{K}\sum _k{x}_{ijk}^p,x_{ij}^q=\frac{1}{K}\sum _k{x}_{ijk}^q,x_{ij}^u=\underset{k}{\max }{x}_{ijk}^u;{\tilde{W}}_j=(w_j^l,w_j^p,w_j^q,w_j^u)analogously$$

6. 6. Construct the fuzzy decision matrix Ũ and the fuzzy weight vector W̃.

   $$\tilde{U}=[{\tilde{x}}_{ij}{]}_{m\times n},\tilde{W}=[{\tilde{W}}_1,{\tilde{W}}_2,\dots ,{\tilde{W}}_n]$$

7. 7. Linear-scale normalization to a comparable scale. Benefit criteria (Ω_B): r̃_ij = (l/d_j*, p/d_j*, q/d_j*, u/d_j*) with d_j* = max_i x_ij^u. Cost criteria (Ω_C): r̃_ij = (a_j^-/u, a_j^-/q, a_j^-/p, a_j^-/l) with a_j^- = min_i x_ij^l.

   $$j\in {\Omega }_B:{\tilde{r}}_{ij}=(\frac{x_{ij}^l}{d_j^{\ast }},\frac{x_{ij}^p}{d_j^{\ast }},\frac{x_{ij}^q}{d_j^{\ast }},\frac{x_{ij}^u}{d_j^{\ast }}),d_j^{\ast }=\underset{i}{\max }{x}_{ij}^u;j\in {\Omega }_C:{\tilde{r}}_{ij}=(\frac{a_j^{-}}{x_{ij}^u},\frac{a_j^{-}}{x_{ij}^q},\frac{a_j^{-}}{x_{ij}^p},\frac{a_j^{-}}{x_{ij}^l}),a_j^{-}=\underset{i}{\min }{x}_{ij}^l$$

8. 8. Construct the weighted normalized fuzzy decision matrix Ṽ = [ṽ_ij] with ṽ_ij = W̃_j (·) r̃_ij (fuzzy product).

   $${\tilde{v}}_{ij}=(v_{ij}^l,v_{ij}^p,v_{ij}^q,v_{ij}^u)=(w_j^l{x}_{ij}^l,w_j^p{x}_{ij}^p,w_j^q{x}_{ij}^q,w_j^u{x}_{ij}^u)$$

9. 9. For each pair (g, f) and criterion j, compute the Hamming distances d(max(ṽ_gj, ṽ_fj), ṽ_gj) and d(max(ṽ_gj, ṽ_fj), ṽ_fj). Comparison rule: ṽ_gj ≥ ṽ_fj iff d(max, ṽ_fj) ≥ d(max, ṽ_gj).

   $$d(\tilde{A},\tilde{B})={\int }_R|{\mu }_{\tilde{A}}(x)-{\mu }_{\tilde{B}}(x)|dx$$

10. 10. Construct the concordance matrix C̃. For each ordered pair (g, f), J_C = {j : ṽ_gj ≥ ṽ_fj} and c̃_gf = Σ_{j∈J_C} W̃_j (fuzzy sum of weights in the concordance set).

    $${\tilde{c}}_{gf}=(c_{gf}^l,c_{gf}^p,c_{gf}^q,c_{gf}^u)=(\sum _{j\in J_C}{w}_j^l,\sum _{j\in J_C}{w}_j^p,\sum _{j\in J_C}{w}_j^q,\sum _{j\in J_C}{w}_j^u)$$

11. 11. Construct the discordance matrix D. d_gf is the ratio of the maximum Hamming distance over discordance criteria to the maximum Hamming distance over all criteria (yielding a crisp scalar in [0,1]).

    $$d_{gf}=\frac{{\max }_{j\in J_D}d(\max ({\tilde{v}}_{gj},{\tilde{v}}_{fj}),{\tilde{v}}_{fj})}{{\max }_{j}d(\max ({\tilde{v}}_{gj},{\tilde{v}}_{fj}),{\tilde{v}}_{fj})},J_D=\{j:{\tilde{v}}_{gj}<{\tilde{v}}_{fj}\}$$

12. 12. Construct Boolean matrix B from the average concordance level C̄ = (c^l, c^p, c^q, c^u). b_gf = 1 iff c̃_gf ≥ C̄ (compared by Hamming distance, since both are TrFNs); else b_gf = 0.

    $$\bar{C}=(c^l,c^p,c^q,c^u),c^{\ast }=\frac{1}{m(m-1)}\sum _{f=1}^m\sum _{g=1}^m{c}_{gf}^{\ast }\text{for }\ast \in \{l,p,q,u\};b_{gf}=1[{\tilde{c}}_{gf}⪰\bar{C}]$$

13. 13. Construct Boolean matrix H from the average discordance level D̄. h_gf = 1 iff d_gf < D̄; else 0.

    $$\bar{D}=\frac{1}{m(m-1)}\sum _{f=1}^m\sum _{g=1}^m{d}_{gf};h_{gf}=1[d_{gf}<\bar{D}]$$

14. 14. Global matrix Z = B ⊗ H (Hadamard product, z_gf = b_gf · h_gf). Build the outranking decision graph G = (V, J): vertex per alternative; an arc A_g → A_f exists iff z_gf = 1. Pairs with z_gf = z_fg = 1 are indifferent; with z_gf = 0 = z_fg are incomparable.

    $$Z=B\otimes H,z_{gf}=b_{gf}\cdot h_{gf};A_{g}S{A}_f\iff z_{gf}=1$$